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A210725 Triangle read by rows: T(n,k) = number of forests of labeled rooted trees with n nodes and height at most k (n>=1, 0<=k<=n-1).

%I A210725 #33 Apr 26 2021 11:19:44
%S A210725 1,1,3,1,10,16,1,41,101,125,1,196,756,1176,1296,1,1057,6607,12847,
%T A210725 16087,16807,1,6322,65794,160504,229384,257104,262144,1,41393,733833,
%U A210725 2261289,3687609,4480569,4742649,4782969,1,293608,9046648,35464816,66025360,87238720,96915520,99637120,100000000
%N A210725 Triangle read by rows: T(n,k) = number of forests of labeled rooted trees with n nodes and height at most k (n>=1, 0<=k<=n-1).
%H A210725 Alois P. Heinz, <a href="/A210725/b210725.txt">Rows n = 1..141, flattened</a>
%H A210725 J. Riordan, <a href="http://dx.doi.org/10.1016/S0021-9800(68)80033-X">Forests of labeled trees</a>, J. Combin. Theory, 5 (1968), 90-103.
%e A210725 Triangle begins:
%e A210725   1;
%e A210725   1,    3;
%e A210725   1,   10,   16;
%e A210725   1,   41,  101,   125;
%e A210725   1,  196,  756,  1176,  1296;
%e A210725   1, 1057, 6607, 12847, 16087, 16807;
%e A210725   ...
%p A210725 f:= proc(k) f(k):= `if`(k<0, 1, exp(x*f(k-1))) end:
%p A210725 T:= (n, k)-> coeff(series(f(k), x, n+1), x, n) *n!:
%p A210725 seq(seq(T(n, k), k=0..n-1), n=1..9); # _Alois P. Heinz_, May 30 2012
%p A210725 # second Maple program:
%p A210725 T:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
%p A210725       binomial(n-1, j-1)*j*T(j-1, h-1)*T(n-j, h), j=1..n))
%p A210725     end:
%p A210725 seq(seq(T(n, k), k=0..n-1), n=1..10);  # _Alois P. Heinz_, Aug 21 2017
%t A210725 f[_?Negative] = 1; f[k_] := Exp[x*f[k-1]]; t[n_, k_] := Coefficient[Series[f[k], {x, 0, n+1}], x, n]*n!; Table[Table[t[n, k], {k, 0, n-1}], {n, 1, 9}] // Flatten (* _Jean-François Alcover_, Oct 30 2013, after Maple *)
%o A210725 (Python)
%o A210725 from sympy.core.cache import cacheit
%o A210725 from sympy import binomial
%o A210725 @cacheit
%o A210725 def T(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*T(j - 1, h - 1)*T(n - j, h) for j in range(1, n + 1)])
%o A210725 for n in range(1, 11): print([T(n, k) for k in range(n)]) # _Indranil Ghosh_, Aug 21 2017, after second Maple code
%Y A210725 Diagonals include A000248, A000949, A000950, A000951, A000272.
%K A210725 nonn,tabl
%O A210725 1,3
%A A210725 _N. J. A. Sloane_, May 09 2012